Finite Group Databasebeta

← #26209

Subgroups — #26209

Order 256.

415 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2414C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic, complemented
3#2099C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#1926C211yesmaximal, characteristic, complemented
8#1921C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1825C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32#394—21no—
33#392—21no—
34#390—21no—
35#390—21no—
36#390C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#388—21no—
41#384C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4—21no—
46C4×C4⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4D411yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#121—22no—
73#121—22no—
74#121—22no—
75#121—41no—
76C4×D4—22no—
77C4×D4—22no—
78#117—22no—
79#117—22no—
80#117—22no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112#107—22no—
113#106—41no—
114#102—22no—
115#102—41no—
116#102—41no—
117#102—41no—
118C22⋊C8—22no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C24C2×D411yescharacteristic
128C24C2×D411yescharacteristic
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—41no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—81no—
142C2×D4—81no—
143C2×D4—81no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4#4112yes—
166C22×C4#4112yes—
167C22×C4C2411yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—44no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4#4112yes—
198C4⋊C4#4112yes—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42—42no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23#12312yes—
238C23C22×D411yescharacteristic
239D4—44no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—84no—
245D4—84no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4#12312yes—
286C2×C4#13512yes—
287C2×C4C2×#4111yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4H(5,𝔽2)11yescharacteristic
292C2×C4H(5,𝔽2)11yescharacteristic
293C8—41no—
294C8—41no—
295C8—41no—
296C8—41no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22C2×#12311yescharacteristic
319C22#54611yescharacteristic
320C22#54611yescharacteristic
321C22C2×#13511yescharacteristic
322C22#57511yescharacteristic
323C22#57611yescharacteristic
324C22#57711yescharacteristic
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—42no—
330C4—42no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4#52911yescharacteristic
337C4#53811yescharacteristic
338C4#55311yescharacteristic
339C4#55311yescharacteristic
340C2—42no—
341C2—81no—
342C2—161no—
343C2#244411yescharacteristic
344C2#309111yescharacteristic
345C2#309611yescharacteristic
346C2#330411yescharacteristic
347C2#330511yescharacteristic
348C2#330711yescharacteristic
349C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2414C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2354C2semidirect
3. #2099C2semidirect
4. #2077C2semidirect
5. #2071C2semidirect
6. #2033C2semidirect
7. #1926C2semidirect
8. #1921C2semidirect
9. #1893C2semidirect
10. #1875C2semidirect
11. #1875C2semidirect
12. #1869C2semidirect
13. #1854C2semidirect
14. #1825C2semidirect
15. #1735C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#24142
#23542
#20992
#20772
#20712
#20332
#19262
#19212
#18932
#18752
#18692
#18542
#18252
#17352